Sadly most if not all VPS's are hard to transfer since the backup images are specific to their hosting, so even if they have daily backups images these are for recovering the VPS on their system.
There is no simple way to copy a image from one VPS to another due to the settings on the opposite VPS host and in fact most VPS don't even have a image import feature to allow you to do this, and if you was to do it via SSH it would more than likely break the new VPS.
It's also worth mentioning not only configurations vary from VPS to VPS, but the hardware and VM software used with be different, and most have in house alterations to them. There is OpenVZ, Xen, vServer, KVM, Cloud VPS and so forth each will be altered to their settings.
The best way forward would be to backup your configuration files and then import those settings into the new server while correcting things like IP Addresses along the way.
Since you haven't mentioned what is running on the server I can't advise exactly which files you should backup, but generally its the vhosts, apache2, host file, and so forth.
I understand that you had someone set the VPS up for you but it'll require someone to setup the new one too as mentioned there's no simple way of doing it. But the fact your on here asking is a good sign that you are capable of finding out how this come be done.
In your position I'd write down exactly what your using, IE Apache2, Fail2Ban etc etc, and write down exactly what domains and so on you have on this server and from there you should be able to work out what needs to be done on the new server and what configuration files you need to take a look at. I strongly disadvise just copying the old configs to the new without checking them since some can have server-side configs which again will break the new.